Lesson 6.7 Lesson 6.7 Circumference and Arc LengthCircumference and Arc Length

Objectives/Assignment

• Find the circumference of a circle and the length of a circular arc.

• Use circumference and arc length to solve real-life problems.

• Homework:– Lesson 6.7/1-11, 17, 19, 22

• Quiz Wednesday

• Chapter 6 Test Friday

Finding Circumference and Arc Length

• The circumference of a circle is the distance around the circle.

• For all circles, the ratio of the circumference to the diameter is the same: or pi.

• The exact value of Pi = • The approximate value of Pi ≈ 3.14

Distance around the circle

Minor ArcMinor Arc•Use 2 lettersUse 2 letters•Angle is less than or equal to 180Angle is less than or equal to 180

XX YY

ZZ

120°120°99

XYZ Major ArcMajor Arc•Use 3 lettersUse 3 letters•Angle is greater than Angle is greater than 180180

XZ

m XZ = m<XCZ = 120o

C

Central Angle: Any angle whose vertex is the center of the circle

Terminology

m XYZ = m<XCZ = 240o

Circumference of a Circle

• The circumference C of a circle is

• C = d or C = 2r, where

• d is the diameter of the circle and

• r is the radius of the circle (2r = d)

diameter d

Comparing Circumferences

• Tire Revolutions• Tires from two different

automobiles are shown.

• How many revolutions does each tire make while traveling 100 feet?

Tire A Tire B

• C = d

• diameter = 14 + 2(5.1) d = 24.2 inches

• circumference = (24.2)

• C ≈ 75.99 inches.

Comparing Circumferences - Tire A

• C = d

• diameter = 15 + 2(5.25)

• d = 25.5 inches

• Circumference = (25.5)

• C ≈ 80.07 inches

Comparing Circumferences - Tire B

• Divide the distance traveled by the tire circumference to find the number of revolutions made.

• First, convert 100 feet to 1200 inches.

TIRE A: 100 ft.75.99 in.

1200 in.75.99 in.= 100 ft.

80.07 in.1200 in.

80.07 in.=

15.8 revolutions

TIRE B:

14.99 revolutions

Comparing CircumferencesTire A vs. Tire B

Revolutions = distance traveled circumference

COMPARISON: Tire A required more revolutions to cover the same distance as Tire B.

Arc Length• The length of part of the circumference.

The length of the arc depends on what two things?

1) The measure of the arc.

2) The size of the circle (radius).

An arc length measures distance while the measure of an arc is in degrees.

Portions of a Circle: Determine the Arc measure based on the portion given.

A. B. C. D.

¼ of a circle: ½ of a circle: 1/3 of circumference : 6π out of a total 36π on the circle: ¼ ● 360

90o

90o

½ ● 360

180o

180o

1/3 ● 360

120o

120o

1/6 ● 360

60o

60o

90o 180o

120o

60o

An arc length is a portion of the circumference of a circle.

Arc Length Formula

m˚360˚ 2πrArc Length =

measure of the central angle or arc

The fraction of the circle!

The circumference of the entire circle!

.

Arc Length• In a circle, the ratio of the

length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360°.

Arc length of =360°

• 2r AB

mAB

Arc length linear units (inches/feet/meters …)Arc measure degrees

Arc measure

Finding Arc Lengths

• Find the length of each arc.

5 cm

B

A

50°

a.7 cm

D

C

50°

b. 7 cm

F

E

100°c.

Finding Arc Lengths, con’t.

• Find the length of each arc.

5 cm

B

A

50°

a.

a. Arc length of = AB 50°

360°• 2(5)

a. Arc length of = AB # of °

360°• 2r

4.36 centimetersArc length of

AB

7 cm

D

C

50°

b. b. Arc length of = CD # of °

360°• 2r

b. Arc length of = CD 50°

360°• 2(7)

6.11 centimeters

Finding Arc Lengths, con’t.

• Find the length of each arc.

Arc length of CD

In parts (a) and (b), note that the arcs have the same measure but different lengths because the circumferences of the circles are not equal.

7 cm

F

E

100°

c.c. Arc length of =

# of °

360°• 2r

c. Arc length of = EF 100°

360°• 2(7)

EF

12.22 centimeters

Finding Arc Lengths, con’t.

• Find the length of each arc.

Arc length of EF

Find the exact length of AB

90

6

m AOB

radius

240

12

m AOB

radius

300

12

m AOB

radius

120

2.4

m AOB

radius

108

10 2

m AOB

radius

Fraction of circle:

90o

90o

6

Fraction ● circumference

¼ ● 12π

3π units

240o

240o

12

Fraction of circle:

Fraction ● circumference

2/3 ● 24π

16π units

300o

300o 12

Fraction of circle:

5/6 ● 24π

20π units

A

AB

B

O OAO

B

120o

120o

2.4AO

B

Fraction of circle:

1/3 ● 4.8π

1.6π units

108o

108o

10√2A

B

O

Fraction of circle:

3/10 ● 20√2π

6√2π units

3

2

360

240

4

1

360

90

6

5

360

300

3

1

360

120

10

3

360

108

3.82 m

60º50º5c

m

dmAB

BarclengthA 360

52360

50 arclength

38.1arclength

d360

6082.3

cmarclength 36.4

d6

182.3

Cdm 92.22

7.64 in.

18 in.

Z

Y

X

Using Arc Lengths• Find the indicated measure.

XY

b. mXY

Arc length of 2r= 360°

18

(15.28)=

135° m

XY

XY

m360° •

XY

Substitute and Solve for XY

m

m

18 in. 2(7.64)= 360°

XY

m

Finding Arc Length• Race Track. The track shown has six lanes. • Each lane is 1.25 meters wide. • There is 180° arc at the end of each track. • The radii for the arcs in the first two lanes are

given. a. Find the distance around Lane 1. (use r1)

b. Find the distance around Lane 2. (use r2)

a. Find the distance around Lanes 1 and 2. The track is made up of

two semicircles two straight sections with length s

Finding Arc Length, con’t

• Distance = 2s + 2r1

= 2(108.9) + 2(29.00)

400.0 meters

• Distance = 2s + 2r2

= 2(108.9) + 2(30.25)

407.9 meters

Lane 1

Lane 2

Finding Arc Length, con’t

Find each arc length. Give answers in terms of and rounded to the nearest hundredth.

Finding Arc Length

FG

Use formula for area of sector.

5.96 cm 18.71 cm

Substitute 8 for r and 134 for m.

Simplify.

Find each arc length. Give answers in terms of and rounded to the nearest hundredth.

Finding Arc Length

an arc with measure 62 in a circle with radius 2 m

Use formula for area of sector.

0.69 m 2.16 m

Substitute 2 for r and 62 for m.

Simplify.

Check It Out!

Find each arc length. Give your answer in terms of and rounded to the nearest hundredth.

GH

Use formula for area of sector.

Substitute 6 for r and 40 for m.

Simplify.= m 4.19 m

Check It Out!

Find each arc length. Give your answer in terms of and rounded to the nearest hundredth.

an arc with measure 135° in a circle with radius 4 cm

Use formula for area of sector.

= 3 cm 9.42 cm

Substitute 4 for r and 135 for m.

Simplify.

Upcoming

• 6.7 Monday• 6.7 Tuesday• Chapter Review Wednesday• Chapter Review Thursday• Chapter 6 Test Friday